My chest on the left side hurts when I cough I was coughing really hard and on the left side I heard a loud pop. Rite at the rib cage. I can breath. But if itry to cough it hurts really bad just woundering could it be a pulled muscle or something

Injured rib. You may have injured ribs, in addition to muscle soreness. Severe coughing may cause this. You will want to get x-rays asap to make a better diagnosis.
Yes . Yes musculoskeletal pain is most common cause of chest pain especially given your history of coughing it is likely that this has resulted in muscle strain. It is most appropriately treated by suppressing cough to avoid further muscle strain.
Yes. Coughing causes the rib cage muscles and respiratory diaphram to contract very hard. You could have strained a chest muscle or even injured a rib cartilage. Try ice, alleve. See your family doc if the cough persists.